Camera: Keep perseptive by changing focal length?
If

d1 is the distance to the object viewed by the camera with a focal length f1
d2 is the wanted distance when the focal length is f2

Then:
d1 / f1 = d2 / f2

So that:
d2 = f2 * d1 / f1

If you want to focus on an object, you need to drive the camera on its local Z axis considering this distance shift. So we need to get the distance at the starting point and calculate the new position when the focal length is changed.
A way to do it in Blender is to use drivers.
As we want to keep the original camera position (which corresponds to d1), we can use delta transforms to do it.
These drivers will use some Python functions:
import bpy
from mathutils import Vector

def calculate_delta(self, initial_focal_length):
    cube_location = bpy.data.objects['Cube'].location
    location = self.location
    distance = (location - cube_location).length
    focal_lens = self.data.lens
    delta = distance - (focal_lens * distance / initial_focal_length)
    return delta 

def in_world(self, delta):
    matrix = self.matrix_world.copy()
    matrix[0][3] = 0
    matrix[1][3] = 0
    matrix[2][3] = 0
    return (matrix @ Vector((0, 0, -1))) * delta

def move_delta(self, initial_focal_length):
    delta = calculate_delta(self, initial_focal_length)
    return in_world(self,delta)

bpy.app.driver_namespace[&quot;move_delta&quot;] = move_delta

The script calculates the delta from the given formula, translates it in world space to shift the camera. And returns the wanted vector.
Then, it binds the function to the driver namespace.
Run this script once in order to use it in drivers.
Attach a driver to each component of the delta location:

The drivers use the function giving self (the camera), the initial focal length (50) and get the corresponding componant: x, y or z.

Note: the cube which is focused on in this example has its origin placed at the front plane.